This invention relates to methods of applying external magnetic fields and Radio Frequency (RF) pulses to fluid saturated porous media and subsequently receiving and analyzing signals therefrom to determine properties of the fluid saturated porous media, and more particularly, to methods which utilize Nuclear Magnetic Resonance (NMR) to analyze properties of subterranean formations and borehole core samples.
NMR instruments are known to be used which employ pulsed RF fields to excite porous media containing fluids in pore spaces thereby inducing signals to be emitted from the fluid and porous media. The emitted signals are then analyzed to determine important properties of the fluid and porous media. Emitted signals of particular value include proton nuclear magnetic resonance signals. These signals are analyzed to provide data including porosity, pore size distribution of the porous media, percentage oil and water content, permeability, fluid viscosity, wettability, etc.
NMR measurements can be done using, for example, the centralized MRIL.RTM. tool made by NUMAR, a Halliburton company, and the sidewall CMR tool made by Schlumberger. The MRIL.RTM. tool is described, for example, in U.S. Pat. No. 4,710,713 to Taicher et al. Details of the structure and the use of the MRIL.RTM. tool, as well as the interpretation of various measurement parameters are also discussed in U.S. Pat. Nos. 4,717,876; 4,717,877; 4,717,878; 5,212,447; 5,280,243; 5,309,098; 5,412,320; 5,517,115, 5,557,200 and 5,696,448. A Schlumberger CMR tool is described, for example, in U.S. Pat. Nos. 5,055,787 and 5,055,788 to Kleinberg et al.
The content of the above patents is hereby expressly incorporated by reference.
Proton nuclear magnetic resonance signals measured from a fluid-saturated rock contains information relating to the bulk and surface relaxation and diffusion coefficients of pore fluids, the pore size distribution, and the internal magnetic field gradient distribution within pore spaces. These multiple pieces of information are often coupled together in a complicated fashion making it very difficult to sort out the value of each of the aforementioned individual physical quantities.
Diffusion may be qualitatively described as the process by which molecules move relative to each other because of their random thermal motion. This diffusive action of molecules enhances the relaxation rate of NMR signals in a magnetic field gradient.
For fluids in rock pores, three independent mechanisms are primarily responsible for the relaxation or decay of magnetic resonance signals (Coates, G. R., Xiao, L. and Prammer, M. G., NMR Logging Principles and Applications, p. 46, (1999)):
bulk fluid relaxation processes, which determine the value for T1 and T2 for bulk fluids;
surface relaxation which affects both T1 and T2; and
diffusion in the presence of magnetic field gradients, which only affects T2 relaxation.
All three processes act in parallel, and the apparent T1 and T2 of pore fluids are given by:                                           1                          T                              1                ,                app                                              =                                    1                              T                                  1                  ⁢                  B                                                      +                          1                              T                                  1                  ⁢                                      xe2x80x83                                    ⁢                  S                                                                    ;                            (        1        )                                                      1                          T                              2                ,                app                                              =                                    1                              T                                  2                  ⁢                  B                                                      +                          1                              T                                  2                  ⁢                                      xe2x80x83                                    ⁢                  S                                                      +                          1                              T                                  2                  ⁢                  D                                                                    ;                            (        2        )                                                      1                          T                              1                ,                                  2                  ⁢                                      xe2x80x83                                    ⁢                  S                                                              =                                    ρ                              1                ,                2                                      ⁢                          S              V                                      ;                            (        3        )                                                      1                          T                              2                ⁢                                  xe2x80x83                                ⁢                D                                              =                                    1              3                        ⁢                          γ              2                        ⁢                          g              2                        ⁢            D            ⁢                          xe2x80x83                        ⁢                          τ              2                                      ;                            (        4        )            
where:
T1,app is the measured apparent longitudinal relaxation time of the pore fluid;
T1B is the longitudinal relaxation time of the pore fluid in bulk phase, i.e., when it is an infinite fluid medium not restricted by pore walls;
T1S is the longitudinal relaxation time of the pore fluid due to the surface relaxation mechanism;
T2,app is the measured apparent transverse relaxation time of the pore fluid;
T2B is the transverse relaxation time of the pore fluid in bulk phase, i.e., when it is an infinite fluid medium not restricted by pore walls;
T2S is the transverse relaxation time of the pore fluid due to the surface relaxation mechanism; and
T2D is the equivalent relaxation time of the pore fluid of the enhanced relaxation rate due to diffusion of spins in a magnetic field gradient;
where xcex3 is the gyromagnetic ratio, xcfx84 is the time between the initial xcfx80/2 pulse and the subsequent xcfx80 pulse, or half the echo spacing in a Carr-Purcell-Meiboom-Gill (CPMG) (Carr, H. Y. and Purcell, E. M., Phys. Rev. 94, 630 (1954) and Meiboom, S. and Gill, D., Rev. Sci. Instrum. 29, 668 (1958)) experiment, xcfx811,2 is the surface relaxivity for T1,2 surface relaxation; S is the area of the pore surface; V is the volume of the pore; g is the magnetic field gradient, and D is the diffusion coefficient of the spins in the fluid.
Thus the measured magnetization decay, i.e., the spin echo amplitude as a function of time ti (the decay time for the i-th echo) for a single pore size system saturated with a single pore fluid can be expressed as:                                                                                           M                  ⁡                                      (                                          t                      i                                        )                                                                    M                  0                                            =                            ⁢                              exp                ⁡                                  [                                                            -                                                                        t                          i                                                                          T                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            B                                                                                                                -                                                                  t                        i                                                                    T                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          S                                                                                      -                                                                  t                        i                                                                    T                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          D                                                                                                      ]                                                                                                        =                            ⁢                                                exp                  ⁡                                      [                                                                  -                                                                              t                            i                                                                                T                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              B                                                                                                                          -                                                                        t                          i                                                                          T                                                      2                            ⁢                                                          xe2x80x83                                                        ⁢                            S                                                                                              -                                                                        1                          3                                                ⁢                                                  γ                          2                                                ⁢                                                  g                          2                                                ⁢                                                  xe2x80x83                                                ⁢                                                  τ                          2                                                ⁢                                                  Dt                          i                                                                                      ]                                                  .                                                                        (        5        )            
Note that Eq.(3) is valid for the fast diffusion limit, i.e., when the diffusion time for a spin to traverse the pore is much shorter than the surface relaxation time. Eq.(4) is strictly valid only for an infinite medium and approximately valid in fluid-saturated porous media when the Gaussian approximation for the phase distribution of spins is satisfied (Dunn, K. J. et al, SPWLA 42nd Annual Symposium, Paper AAA, Houston, Tex., June 17-20 (2001); and Dunn, K. J., Magn. Reson. Imaging, 19, 439, (2000)). In the present discussion, it is assumed that such Gaussian approximation is valid. Deviation of the physical quantities from their expected values may be attributed, in part, to the failure of such assumption.
Natural fluid-saturated rocks generally have multiple pore sizes. If the surface relaxation strength is reasonably strong (i.e., xcfx81xcx9c10 xcexcm/s), the spins in the pore fluid can only diffuse a short distance of a few pores, the spins at each pore relax more or less independently of the spins in other pores in a diffusion decoupled situation.
Thus, the spin echo amplitude as a function of time ti for a multiple pore size system when there is no magnetic field gradient can be expressed as:                                           M            ⁡                          (                              t                i                            )                                            M            0                          =                              ∑            j                    ⁢                                    f              j                        ⁢                          ⅇ                                                -                                      t                    i                                                  /                                  T                                      2                    ⁢                                          xe2x80x83                                        ⁢                    j                                                                                                          (        6        )            
where the first term in the exponent of Eq.(5) is neglected because T2B greater than  greater than T2S, and ti is the decay time for the i-th echo in a CPMG experiment, fj is the volume fraction of the pores characterized by a common T2 relaxation time T2j, and 1/T2j=xcfx812Sj/Vj (where Sj is the pore surface area and Vj is the pore volume of pore size j). T2j=xcex1j/xcfx812 with xcex1j=Vj/Sj as a measure of the pore size.
The LHS of Eq.(6) can be obtained from a CPMG measurement, whereas the volume fractions fj on the RHS of Eq.(6) are to be solved from the data analysis. This problem is usually treated by assuming a set of pre-selected T2j values equally spaced on a logarithmic scale and solving for the amplitude fj associated with T2j. The solution obtained is called the T2 distribution (i.e., fjvs T2j). This mathematical procedure of obtaining the T2 distribution is common in NMR relaxation data analysis and is referred to as an inversion process. Since T2j=xcex1j/xcfx812 in the fast diffusion limit, the T2 distribution frequently reflects the pore size distribution of the rock.
For a fluid-saturated porous medium in a magnetic field gradient, the problem becomes a bit more complicated. The magnetic field inhomogeneities can come from an externally applied field gradient, which is uniform over the pore scale, and/or from local field gradients which have a spatial variation across individual pores. The latter is caused by the magnetic susceptibility contrast between the solid matrix and pore fluids.
If the externally applied magnetic field gradient is much larger than the local field gradients in the pore space due to magnetic susceptibility contrast, the spin echo amplitude can be expressed as a function of time ti for a multiple pore size system as:                                           M            ⁡                          (                              t                i                            )                                            M            0                          =                              ∑            j                    ⁢                                    f              j                        ⁢                          ⅇ                                                -                                      t                    i                                                  /                                  T                                      2                    ⁢                                          xe2x80x83                                        ⁢                    j                                                                        ⁢                                          ⅇ                                                      -                                          γ                      2                                                        ⁢                                      g                    2                                    ⁢                                      xe2x80x83                                    ⁢                                      τ                    2                                    ⁢                                                            Dt                      i                                        /                    3                                                              .                                                          (        7        )            
Note that the enhanced relaxation term due to diffusion, exe2x88x92xcex32g2xcfx842Dti/3, has a fixed field gradient g determined by the externally applied field gradient. It is not related to the pore sizes, and can be pulled out of the summation sign. Thus, this term can be decoupled from the summation over different pore sizes, which makes the analysis relatively easy.
Frequently, the probed zone of an NMR logging tool has a magnetic field gradient distribution, S(g), where the gradient varies over a scale much larger than the pore scale such that the value g can be treated as a constant over the dimension of a representative volume element of the probed zone. In this case, it is the same as Eq. (7), i.e., for each volume element, it has a constant g value. Thus, the enhanced relaxation term due to diffusion can be pulled out of the summation over different pore sizes and can be expressed in the following form:                                           M            ⁡                          (                              t                i                            )                                            M            0                          ⁢                  ∫                                    S              ⁡                              (                g                )                                      ⁢                          ⅇ                                                -                                      γ                    2                                                  ⁢                                  g                  2                                ⁢                                  xe2x80x83                                ⁢                                  τ                  2                                ⁢                                                      Dt                    i                                    /                  3                                                      ⁢                          ⅆ              g                        ⁢                                          ∑                j                            ⁢                                                f                  j                                ⁢                                  ⅇ                                                            -                                              t                        i                                                              /                                          T                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        j                                                                                                                                                    (                  8          ⁢          a                )            
This is not the case, however, when the local field gradients become significant or dominant. These internal field gradients are affected by the pore shapes and sizes, and vary over the pore scale. They cannot be decoupled from the summation over different pore sizes. The spin echo amplitude as a function of time ti is now expressed as:                                                         M              ⁡                              (                                  t                  i                                )                                                    M              0                                =                                    ∑              j                        ⁢                                          f                j                            ⁢                              ⅇ                                                      -                                          t                      i                                                        /                                      T                                          2                      ⁢                                              xe2x80x83                                            ⁢                      j                                                                                  ⁢                                                ∫                  j                                      xe2x80x83                                                  ⁢                                                                            P                      j                                        ⁡                                          (                      g                      )                                                        ⁢                                      ⅇ                                                                  -                                                  γ                          2                                                                    ⁢                                              g                        2                                            ⁢                                              xe2x80x83                                            ⁢                                              τ                        2                                            ⁢                                                                        Dt                          i                                                /                        3                                                                              ⁢                                      ⅆ                    g                                                                                      ⁢                  xe2x80x83                                    (                  8          ⁢          b                )            
where Pj(g) is the volume fraction which has a gradient value of g within the pore of size j, and                                           ∫            j                    ⁢                                                    P                j                            ⁡                              (                g                )                                      ⁢                          xe2x80x83                        ⁢                          ⅆ              g                                      =        1                            (        9        )            
is normalized to 1.
The fact that this enhanced relaxation term due to diffusion cannot be decoupled from the summation presents a problem to the data analysis, namely, information cannot be obtained on the internal field gradient distribution as a function of pore size, and thus corrections cannot be made on its adverse effect on data analysis.
Similar problems arise when information is to be obtained regarding brine and crude oil saturated rocks from regular CPMG measurements, where the internal field gradients are small and are not a concern, but the crude oil also has a T2 distribution which is to be solved. The spin echo amplitude as a function of time ti is now expressed as:                                                                                           M                  ⁡                                      (                                          t                      i                                        )                                                                    M                  0                                            =                            ⁢                                                ∑                  j                                ⁢                                  (                                                                                    f                                                  w                          ,                          j                                                                    ⁢                                              ⅇ                                                                              -                                                          t                              i                                                                                /                                                      T                                                          2                              ⁢                                                              xe2x80x83                                                            ⁢                              j                                                                                                                          ⁢                                              ⅇ                                                                              -                                                          γ                              2                                                                                ⁢                                                      g                            2                                                    ⁢                                                      xe2x80x83                                                    ⁢                                                      τ                            2                                                    ⁢                                                      D                            w                                                    ⁢                                                                                    t                              i                                                        /                            3                                                                                                                +                                                                                                                                        ⁢                                                f                                      o                    ,                    j                                                  ⁢                                  ⅇ                                                            -                                              t                        i                                                              /                                          T                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        j                                                                                            ⁢                                                      ∫                                          xe2x80x83                                                        ⁢                                                            P                      ⁡                                              (                                                  D                          oil                                                )                                                              ⁢                                          ⅇ                                                                        -                                                      γ                            2                                                                          ⁢                                                  g                          2                                                ⁢                                                  xe2x80x83                                                ⁢                                                  τ                          2                                                ⁢                                                  D                          oil                                                ⁢                                                                              t                            i                                                    /                          3                                                                                      ⁢                                          ⅆ                                              D                        oil                                                                                                        )                                                          (        10        )            
where fw,j is the volume fraction for brine and fo,j is the volume fraction for crude oil, and the diffusion coefficient of the crude oil, Doil, has a distribution, P(Doil), and
∫P(Doil)dDoil=1xe2x80x83xe2x80x83(11)
is normalized to one.
Again, in Eq.(10), the enhanced relaxation term due to diffusion for oil cannot be pulled out of the summation because it is related to the summation indices in the following manner:                               D                      oil            ,            j                          =                  a          ⁢                                    T                              K                .                                                    η              j                                                          (        12        )                                                      T                          2              ,              j                                =                      b            ⁢                                          T                                  K                  .                                                            η                j                                                    ⁢                  xe2x80x83                ⁢                  
                ⁢                  xe2x80x83                ⁢                  and          ⁢                      xe2x80x83                    ⁢          thus                                    (        13        )                                          D                      oil            ,            j                          =                              a            b                    ⁢                      T                          2              ,              j                                                          (        14        )            
following the Constituent Viscosity Model (CVM) suggested by Freedman et al. (Freedman, R., Sezginer, A., Flaum, M., Matteson, A., Lo, So., and Hirasaki, G. J., SPE Paper 63214, Society of Petroleum Engineers, Dallas, Tex. (2000)), where a and b are proportional constants, TK is the absolute temperature in Kelvin, and xcex7j is the constituent viscosity related to the chosen T2j and Doil,j,.
Although the enhanced relaxation term due to diffusion for water is not related to the summation indices because the diffusion coefficient for water is a single value and does not have a distribution, it cannot be treated separately. In fact, even though the diffusion coefficient for water is a single value, it can also have an apparent distribution of diffusion coefficients due to the following reasons:
(1) effect of noise which prevents the result from having a sharp and well-defined value,
(2) restricted diffusion effect from different pore sizes, and
(3) unknown internal field gradients which superimpose upon any fixed gradient over the pore dimension and lump their effect on the diffusion coefficient when a fixed gradient value is used in the analysis.
Accordingly, there is a need for one or more methods which uncouple the entangled information due to diffusion effects received during NMR analysis so that determinations can be made regarding properties of porous media and fluids contained within pore spaces from a subterranean formation or from a core sample of rock. Furthermore, there is a need to display these results in a manner that is particularly effective in visualizing the results of the analysis.
The present invention includes a method for analyzing the properties of a porous medium containing fluids. The method includes the following steps. A static magnetic field B0 is applied to a porous medium containing fluid in pore spaces to polarize spins of protons in the porous medium and fluids and create an overall magnetization. A series of differentiated sequences of radio frequency (RF) pulses and possible gradient pulses is applied to the porous medium and fluids contained therein at the resonance frequency of the protons and at a given external magnetic field to excite the magnetization. The pulse sequences have a preparation part followed by a first window of a time length t0 and a second window, wherein the portion of the pulse sequence in the first window of at least one pulse sequence is differentiated by at least one differentiating variable vd from the portion of the pulse sequence in the first window of another of the pulse sequences.
Induced resonance signals are acquired from the porous medium and fluids contained therein during the second windows of the pulse sequences. The resonance signals are acquired as a function of ti, PA1(ti), and vd, where ti is the time of the ith spin echo in the second window as measured from initiation of that second window, PA1(ti) is the spin echo amplitude, and vd is the value of the differentiating variable. The acquired induced resonance signals, acquired as a function ti, PA1(ti), and vd, are processed to determine properties of the porous medium and fluids contained therein as a function of T2, one of g and D, and PA3, where T2 is the transverse relaxation time of the protons, g is the internal field gradient in the pore spaces, D is the diffusion coefficient of the fluids in the pores, and PA3 is an amplitude distribution proportional to proton population. The use of the differentiated pulse sequences in the first windows of the pulse sequences creates differentiated decay amplitudes at the initiation of the second windows. The acquisition of echo signals in the second windows beginning at a time t0 and accommodates the uncoupled determination of the quantities of transverse relaxation time T2 of protons, either the internal field gradients g or the diffusion coefficient D, and the amplitude proportional to proton population PA3.
Ideally, the values T2, one of g and D, and PA3 are plotted on orthogonal axes to produce a 2D NMR plot to provide visualization of properties of the porous medium and fluids contained therein.
Another method for analyzing the properties of a porous medium containing fluids is also disclosed. A static magnetic field B0 is applied to a porous medium containing fluids in pore spaces to polarize spins of protons in the porous medium and fluids and create an overall magnetization. Then, a technique called magic angle spinning is applied to the porous medium followed by a series of rotor synchronized radio frequency (RF) pulses. The RF pulses include a xcex8 pulse followed by a xcfx80/2 pulse at time xcfx841. Free Induction Decay (FID) signals are acquired from the porous medium and fluids contained therein as a function of recovery time xcfx841.
The acquired FID signals are Fourier transformed to obtain a plurality of proton chemical shift spectra at different recovery times xcfx841 and further inverted to obtain an amplitude which is proportional to proton population as a function of proton chemical shift and T1 relaxation time.
Subsequently, the values T1 relaxation time, proton chemical shift and the amplitude proportional to the proton population are plotted on orthogonal axes to produce a 2D NMR plot for visualization of properties characteristic of the porous medium and fluids contained therein.
The present invention further includes storing the above analytical methods on a computer readable media for implementation by NMR instrumentation. Further, the above methods may be practiced on core samples in a laboratory. The determination of internal field gradient and/or diffusion coefficient distribution may be made as part of a downhole logging operation in a wellbore.
It is an object of the present invention to provide novel pulse sequences which allow diffusion effects, due to significant internal field gradients found in pore spaces of porous media, to be decoupled from overall T2 relaxation times.
It is another object to provide a method for determining diffusion coefficient distribution as a function of T2 relaxation time in a porous medium containing fluids.
It is yet another object to provide a novel series of pulse sequences wherein each pulse sequences includes a preparation part, a first window and a second window. The portion of the pulse sequences in the first windows of the series of pulse sequences are differentiated from one another so that the spin echo amplitudes are differentiated from each other at the initiation of each of the second windows thereby uncoupling the diffusion effect from the T2 relaxation time.
It is a further object of the present invention to provide a mathematical method for handling the data contained in the attenuated T2 amplitude due to the diffusion effect, and to process such information into a useful multi-dimensional representation for petrophysical analysis.
It is yet another object to provide a method which removes or minimizes the effects of the internal magnetic field distribution in porous media, such as rocks, in the analysis of the proton NMR relaxation data obtained from borehole logging measurements to thereby extract information about the properties of the pore fluids saturating the porous media from a proton NMR signal distorted by the presence of internal field gradients and molecular diffusion.